How to turn off mirror to Image texture in UV editor
You can't have custom texture coordinates for both sides of a mirrored object, since they - as well as the geometry are created by the modifier. 
You can however choose how to mirror the UV islands in the modifier's settings, using the checkboxes for flip U and flip V. They should about get you there.
If you need to apply completely idependent UVs, you will have to apply the modifier. If you want to texture paint, and issues with the modifier arise, you can safely duplicate your object, apply the modifier on the copy and paint on that one, deleting it once you're done.